Marwan Mokhtar Performance of a 100 kW Concentrated Solar … · 2014-05-16 · Masdar Institute of Science and Technology, Abu Dhabi 54224, UAE e-mail: [email protected] Matteo

Marwan MokhtarResearch Engineer

Laboratory for Energy and Nano Science (LENS),

Department of Mechanical Engineering,

Masdar Institute of Science and Technology,

Abu Dhabi 54224, UAE

e-mail: [email protected]

Steven A. MeyersResearch Engineer





Peter R. ArmstrongAssociate Professor






Matteo ChiesaAssociate Professor






Performance of a 100 kWth

Concentrated Solar Beam-DownOptical ExperimentAn analysis of the beam down optical experiment (BDOE) performance with full concen-tration is presented. The analysis is based on radiation flux distribution data taken onMar. 21st, 2011 using an optical-thermal flux measurement system. A hypothetical ther-mal receiver design is used in conjunction with the experimental data to determine theoptimal receiver aperture size as a function of receiver losses and flux distribution. Theoverall output of the plant is calculated for various operating temperatures and three dif-ferent control strategies namely, constant mass flow of the heat transfer fluid (HTF), con-stant outlet fluid temperature and real-time optimal outlet fluid temperature. It was foundthat the optimal receiver aperture size (radius) of the receiver ranged between (1.06 and1.71 m) depending on temperature. The optical efficiency of the BDOE ranged from 32%to 37% as a daily average (average over the ten sunshine hours). The daily averagemean flux density ranged between 9.422 kW/m2 for the 1.71 m-receiver and 20.9 kW/m2

for the 1.06 m-receiver. Depending on the control parameters and assuming an openreceiver with solar absorptivity of 0.95 and longwave emissivity of 0.10. The averagereceiver efficiency varied from 71% at 300 �C down to 68% at 600 �C. The overall dailyaverage thermal efficiency of the plant was between 28% and 24%, respectively for theaforementioned temperatures. The peak of useful power collected in the HTF was around105 kWth at 300 �C mean fluid temperature and 89 kWth at 600 �C.[DOI: 10.1115/1.4027576]

1 Introduction

Central receiver concentrated solar power plants promise higherenergy conversion efficiency and consequently lower levelizedenergy cost (LEC) [1]. It is expected that 48% reduction in towerplant costs will come from technology research and development[2]. The beam down concept provides the opportunity of locatingbulky receivers on ground level in addition to the use of smallerreceiver apertures and a final concentrating stage, which is madepossible by the narrower view angle of concentrated radiation [3].

In order to demonstrate the potential of such configuration andinvestigate the possibility of a scale-up, a pilot 100 kWth BDOEwas constructed in 2009 in Masdar City Abu Dhabi (24.442�N54.617�E), see Fig. 1.

The heliostat field consists of 33 ganged-type heliostats of total-ing 280.5 m2 of reflective area [4]. In this paper, we present opti-cal and thermal performance analysis of the BDOE based onexperimental data collected in Mar. 2011.

Optical performance is quantified by the optical efficiency, whichincludes the effects of cosine loss, reflexion loss, beam attenuationloss, blocking, and shading loss in addition to light spillage aroundcentral reflector (CR) mirrors and around the receiver aperture (inter-cept factor).The combined effect of the aforementioned (except forthe final intercept factor) is evaluated experimentally using the fluxmeasurement system. The intercept factor is a function of receiveraperture and flux distribution. Consequently, the receiver aperturesize must be specified in order to evaluate the optical efficiency.

Thermal performance depends on the thermal efficiency of thereceiver, which is also a function of receiver aperture size/geome-try. Thermal performance depends additionally on the receiveroperating temperature.

Due to the inverse relationship of receiver size to optical andthermal efficiency, it is necessary to conduct an optimization,which is based on operating temperature and maximum dailyenergy collection.

2 Experimental Setup

2.1 Heliostat Field. The BDOE heliostat field is designed forresearch convenience, not to be implemented on the utility scale.The heliostats are arranged in three concentric rings designated byletters with the closest ring called ring (A) and the furthest one(C) as depicted in Fig. 2. The numbers attached to these letters onthe drawing increase in an anticlockwise manner with respect tothe south of the field (pointing toward the top of the page). Notethat four heliostats are missing from the each sector, one from theA-ring and three from the C-ring. This explains why there are 45rather than 33 secondary mirrors on the CR.

Each heliostat is comprised of 43 flat facets—canted at differ-ent angles to create a Fresnel reflector such that the rays incidenton the center of each facet will intersect (were the central reflec-tors absent) at the upper focal point of the CR. The area per helio-stat is 8.5 m2 yielding a total flat heliostat mirror area of 280.5 m2

for the entire field. These unconventional heliostats allow for abetter utilization of land space and a more compact system designcompared to conventional heliostats (see Fig. 1(a)).

2.2 Central Reflector. The central tower is 16.0 m in height,supporting 45 fixed, flat CR mirrors. Each of the 33 CR mirrorscurrently in use reflects the converging rays from one heliostatback down onto the target (Table 1).

The CR mirrors are arranged in concentric circles correspond-ing to the heliostat rings. The mirrors in each ring are tilted withrespect to the normal of the target by the same angle, creating ahyperboloidal Fresnel reflector. The innermost ring corresponds tothe heliostats’ A-ring, and has the smallest tilt angle of the three,

Contributed by the Solar Energy Division of ASME for publication in theJOURNAL OF SOLAR ENERGY ENGINEERING: INCLUDING WIND ENERGY AND BUILDING

ENERGY CONSERVATION. Manuscript received April 7, 2014; final manuscript receivedApril 10, 2014; published online May 15, 2014. Editor: Gilles Flamant.

Journal of Solar Energy Engineering NOVEMBER 2014, Vol. 136 / 041007-1Copyright VC 2014 by ASME

Downloaded From: http://solarenergyengineering.asmedigitalcollection.asme.org/ on 05/16/2014 Terms of Use: http://asme.org/terms

whereas the ring located furthermost from the center of the towercorresponds to the C-ring heliostats (Fig. 2).

The CR mirrors in the BDOE plant are exposed to about 10suns. Use of a multifaceted CR reduces temperature relatedmechanical stress and improves natural cooling, thus avoiding thethermal stress problem. In addition, using separate reflectors asopposed to a continuous monolithic hyperboloid surface reducesmanufacturing, shipping, and erection costs.

2.3 Measurement System. To study energy flux distributionand its magnitude on the receiver plane of the BDOE, the concen-trated solar radiation is intercepted by a 5 m� 5 m ceramic tile tar-get, located 2.3 m above ground level (see Fig. 1). These whitetiles can withstand the high flux levels and are highly reflective ina diffuse manner approximating a Lambertian target. This allowsa CCD camera located at the top of the tower to measure the dis-tribution of luminous intensity. Embedded within the tiles at eightlocations are heat flux sensors (HFS) to measure the concentratedsolar flux. The HFSs only provide discrete thermal flux data and

the CCD camera provides data of the luminous intensity on thetarget. A correlation was developed between the flux sensor dataand the CCD camera data so that an accurate measurement of fluxdistribution can be obtained. In this way enough information offlux distribution on the focal plane can be acquired at eachmoment of time which allows the study of the BDOE performanceunder various conditions.

The CCD camera temperature is controlled by a combination ofheating and cooling devices. A proportional integral differentialcontroller regulates the air pressure supplied to a vortex cooler inorder to keep the CCD camera temperature constant at 23 �C.

Fig. 1 BDOE overview. (a) Ganged-Type heliostat field and CR mirrors. CCD cameraaperture is in the center of the CR taking images of the target below it. (b) Embedded withinthe target are water-cooled HFS at eight locations to calibrate the CCD camera images. (c) Atypical luminance map taken by the CCD camera.

Table 1 Summary of BDOE parameters

Item Value

Nominal power 100.0 kWth

Number of heliostats 33Reflective area per heliostat 8.5 m2

Primary reflective area 280.5 m2

CR mirrors reflectivity 95%Heliostats mirrors reflectivity 80%Upper focal point 20.0 mLower focal point 2.0 mCR height 16.0 mCR ring A diameter 3.5 mCR ring B diameter 5.4 mCR ring C diameter 7.4 mHeliostat ring A diameter 17.48 mHeliostat ring B diameter 27.08 mHeliostat ring C diameter 36.68 m

Fig. 2 BDOE heliostat field

041007-2 / Vol. 136, NOVEMBER 2014 Transactions of the ASME


The heat flux sensors used in the measurement system alsorequire cooling. For accurate heat flux measurement, it is moreimportant to prevent transients in water temperature than to keepa precise set point. Cooling water is supplied from a tank of suffi-cient thermal mass to prevent any abrupt change in the water tem-perature and to operate for several hours with less than 20 Ktemperature rise. The water temperature going into the sensorsand the outlet temperature of each sensor are recorded and thenused in correcting the HFS for thermal losses using a correctionmodel developed in Ref. [5].

The foregoing combinations of optical and thermal radiometryprovide the required flux maps. While heat flux measurement isaccurate and reliable, it is only feasible to be implemented at dis-crete points on the target. Optical methods on the other handare less accurate but provide very high spatial resolution. Heatflux measurement is used as a reference for calibrating theoptical system and hence combining the advantages of the twomethods. Similar hybrid measurement systems have been used tomeasure heat flux on the targets of parabolic dishes [6] and towerplants [7,8].

3 Performance Analysis

In this section, we present the method used for analyzing theperformance of the BDOE. The optical performance of the plant(not including the intercept factor) is evaluated from the experi-mental results acquired by a flux measurement system. Then, theintercept factor and the receiver thermal performance are eval-uated simultaneously for varying operating temperatures using theaperture size optimization algorithm which is described later inSec. 3.2.

3.1 Receiver Thermal Performance. Heat loss from solarreceivers is a subject of much literature. McDonald [9] presenteda comprehensive investigation of heat loss from cavity receiversincluding convection, radiation and conduction. Receiver convec-tion loss has been studied in great detail by Clausing [10–12] andothers [13–16]. These studies included analytical and numericalmodels supported by experimental studies for various geometriesand receiver orientations. Most of these studies only consideredlosses from downward facing receivers, Leibfried and Ortjohann[13] also considered upward facing receivers.

Although some studies considered wind induced forced convec-tion [17], most studies neglected forced convection loss from cav-ity receivers [10,13,14,16].

In this paper, we base our analysis on the performance of anupward facing open receiver and neglect forced convection effectsbecause the mean wind speed during the test was small (2.6 m/s).

Here, we present the equations and procedure used in assessingthe performance of a hypothetical receiver. The method is basedon the Hottel-Whillier equation [18] as presented in Duffie andBeckman [19]. The HTF selected for the analysis is liquid sodium.Under steady state conditions the net useful power output of thereceiver is the difference between the absorbed solar radiation andthe thermal loss from the receiver surface.

_Qu ¼ Aða _qin � ULðTmp � TeÞÞ (1)

where _Qu is the useful heat output, a is the absorptivity of thereceiver in the solar spectrum (taken as 95%), _Qin is the solarpower intercepted by the receiver aperture, A, UL is the overallheat loss coefficient, Tmp is the mean plate temperature, and Te isthe air temperature both in K.

Equation (1) is reformulated in terms of the inlet fluid tempera-ture (Tfi) and the heat removal factor (FR). FR is introduced toaccount for the underestimated losses calculated based on the inletfluid temperature (Tfi) instead of mean plate temperature.

_Qu ¼ AFRða _qin � ULðTfi � TeÞÞ (2)

The heat removal factor FR is the ratio of useful gain to what itwould have been had the whole absorber plate been at the inletfluid temperature [19]. FR is given by

FR ¼_mcp

AUL1� e

� AULF0

_mCp

� �(3)

where _m is the mass flow rate, Cp is the specific heat of HTF, andF0 is the receiver plate efficiency factor [19] that accounts for thethermal resistance between the plate and the fluid inside the tubes

F0 ¼ 1

ULDO1

ULDiþ 1

pDihf

� � (4)

where Do is the outer tube diameter, Di is the inner tube diameter,and hf is the convection heat transfer coefficient between the innerwall and the fluid given by

hf ¼Nuf kf

Di(5)

For liquid metals like liquid sodium, a convection heat transfercorrelation between the fluid and the tube walls is given by [20]

Nuf ¼ 6:3þ 0:0167R0:85e Pr0:93 Turbulent

Nuf ¼ 4:36 Laminar(6)

where Pr is the Prandtl number, which is the ratio between kine-matic viscosity and thermal diffusivity Pr¼ �/a, and Re is theReynolds number which represents the ratio between inertial andviscous forces Re¼ vD/�.

The overall heat loss coefficient UL accounts for all heat lossesfrom the receiver. There are five main loss mechanisms associatedwith the receiver of a concentrated solar plant: convective loss,radiative loss, conductive loss, loss due to reflection, and spillage.Conductive heat losses are assumed to be negligible compared toother loss mechanisms, while losses due to spillage are accountedfor in the calculation of radiant power intercepted by the receiver(Sec. 3.2). Reflective losses and spillage around the CR areaccounted for by the fact that radiation is measured at receiveraperture level.

According to Ref. [21], in a well designed receiver, thermallosses account for 5–15% of incident, hence a reasonable estimateusing Tmp is adequate to determine receiver performance.

We assume, as mentioned earlier, that the dominant convectionmode is free convection and that forced convection is negligible.The aperture Nusselt number for laminar and turbulent flows aregiven by [20]

Nunatural ¼ 0:54ðGrLPrÞ1=4Turbulent

Nunatural ¼ 0:14ðGrLPrÞ1=3Laminar

(7)

where Grashof number for free convection is given by [20]

GrL ¼bDTgL3

�2(8)

where b is the volumetric thermal expansion coefficient in K�1,DT is the temperature difference between mean plate temperature(Tmp), and ambient air temperature (Te) in K, g is the gravitationalacceleration in m/s2, L¼ 2 R is the characteristic length of thereceiver equal to its diameter in m, and � is the kinematic viscos-ity of air in m2/s.

Hence the convection heat transfer coefficient (in W/m2K) isgiven by

Journal of Solar Energy Engineering NOVEMBER 2014, Vol. 136 / 041007-3


�hc ¼kNu

L(9)

where k is the is the conductivity of air in W/mK and L¼ 2 R isagain the aperture diameter in m.

The temperature dependence of fluid properties was taken intoaccount during calculations.

The radiative heat loss is given by

_Qrad ¼ AerðT4mp � T4

skyÞ (10)

where _Qrad is the radiative heat loss in W, E is the longwave emis-sivity (taken as 10% assuming selectivity to reduce radiativelosses. The selectivity mechanism is assumed to remain stableacross the operating temperature range), Tmp is the mean platetemperature and Tsky is the effective sky temperature both in K. Apyrgeometer was used to measure effective sky temperature dur-ing the experiment. Hence, the linearized radiative heat transfercoefficient relative to air temperature may be defined as

hr ¼ AerT4

mp � T4sky

Tmp � Te(11)

The mean plate temperature Tmp required for solving the previousequations can be calculated by

Tmp ¼ Tfi þQu

AFRULð1� FRÞ (12)

Finally, the overall heat loss coefficient is given by

UL ¼ �hr þ �hc (13)

3.2 Receiver Intercept Factor. The power intercepted by thereceiver will depend on the receiver aperture size. Spillage iscaused by various optical and mechanical errors in the concentra-tion system, in addition to the effect of sunshape on beam widen-ing. Fig. 3 depicts the day-average intercept factor variation withreceiver aperture radius. The receiver intercept factor c is calcu-lated as follows:

c ¼

ðR

0

ð2p

0

Gðr; hÞdhdr

ð10

ð2p

0

Gðr; hÞdhdr

(14)

where R is the receiver intercept factor, R is the receiver apertureradius, and G(r, h) is the flux map defined in polar coordinates.

3.3 Receiver Aperture Sizing. Optimal receiver aperturesize is determined by maximizing the day average useful power inEqn. 2. This is calculated based on the required outlet mean fluidtemperature. An iterative algorithm is used for evaluating thereceiver performance and finding the optimal receiver size.Depending on the temperature/flow control strategy used, thealgorithm is slightly modified. The main calculation steps are:

(1) Calculation of cumulative radial flux distributions whichgive the relation between receiver size (R) and interceptedpower (Qin).

(2) Specification of control strategy and setpoint of controlvariable: mass flow, outlet temperature or optimal outlettemperature.

(3) Selection of a receiver radius (R).(4) Calculation of receiver performance based on input param-

eters: Tfi, Tfo, R, Qin, and ambient conditions.(a) Radiation and convection heat transfer coefficients

(�hr; �hc) are calculated using an initial value for themass flow rate ( _m) and mean plate temperature (Tmp).

(b) Receiver efficiency factor (F0) is calculated.(c) _m is updated using the calculated (F0) and overall heat

loss coefficient (UL).(d) Net useful power (Qu) is updated using the new _m,

where Qu ¼ _mcpðTfo � TfiÞ.(e) Heat removal factor (FR)is updated with the new Qu.(f) Tmp is updated.(g) Steps a-f are repeated until Tmp is accurate enough.

(5) The calculation (4) is repeated for different R and Tfo.

4 Results

In this section, we present the results obtained from the evalua-tion of BDOE performance under full concentration using theexperimental data collected and the receiver model we presentedin Sec. 3.3. Direct normal irradiation (DNI) during the test day isshown in Fig. 4.

4.1 Optical Efficiency. Figure 5 depicts the optical efficiencycalculated based on the optimal receiver size chosen for each fluidoutlet temperature. Optical efficiency represents the amount ofenergy that is intercepted by the receiver aperture normalized bythe product of incident DNI and heliostat area. Optical efficiencyaccounts for all the factors that reduces the amount of concen-trated radiation until it reaches the receiver, but before it gets con-verted into thermal power. Optical efficiency includes cosine

Fig. 3 Day-average intercept factor (c) as a function of receiveraperture radius (R) Fig. 4 DNI during the test day



factor, reflectivity of both heliostats and the CR mirrors, beamattenuation, CR intercept factor (spillage), incident angle modi-fier, blocking and shading of heliostats, and the intercept factor ofthe receiver.

The test was conducted after all heliostat and CR mirrors werecleaned. The reflectivities of representative mirror samples weremeasured in the lab and were spectrally weighted with a 1.5 airmass solar spectrum. This analysis yielded an 80% reflectivity for

the heliostat mirrors and 95% for the CR mirrors. Under normalconditions however, mirrors were left to soil for several days andthe reflectivity of the mirrors was degraded by dirt and sand,accounting for a considerable optical losses. In Fig. 5, it can beseen that the average efficiency of the receiver is varying with out-let temperature from 32% to 37%; this is because the optimalreceiver size changes based on the specified outlet temperature.Since higher temperatures call for smaller receiver apertures, theoptical efficiency is reduced directly by the lower intercept factorfor smaller receiver radii.

Figure 6 depicts the daily variation of intercept factor for differ-ent receiver sizes based on the upper limit of integration, R, in 14.The drop in intercept factor on either side of solar noon stemsmainly from the well-known effects of off-axis aberration (astig-matism) associated with heliostat optics (Fig. 7). The elongatedflux distributions observed in morning and afternoon result inradiation spilling outside the receiver aperture [22].

4.2 Receiver Thermal Efficiency. Receiver thermal effi-ciency is a function of its mean temperature and incident flux perunit aperture size. Figure 8 illustrates as a function of aperture ra-dius, the incident power on the receiver, convection losses, radia-tion losses, and net useful output. The curves are calculated for anoutlet fluid temperature of 400 �C. It can be seen that a maximumfor net power occurs at a certain receiver radius. Figure 9 depictsthe daily variation of receiver thermal efficiency for several outletfluid temperatures, and also indicates that the receiver thermalefficiency at 300 �C (R¼ 1.71 m) is less than that at 400 �C

Fig. 5 Optical efficiency with optimal aperture

Fig. 6 Intercept factor variation during the test day for differentreceiver aperture sizes

Fig. 7 Luminance maps at different times of the day (local time UTC14) shown in cd/m2. x and y axes are in pixels.Aberration is evident in early and late parts of the day which correspond to reduced intercept factor.

Fig. 8 Net power collected as a function of receiver radius.Convection and radiation losses are also shown as a functionof receiver size, Tfo 5 400 �C. Daily average is calculated overthe ten sunshine hours of the test day.



(R¼ 1.27 m). This is because the optimal receiver aperture size at400 �C is larger, which resulted in a higher thermal loss despitethe lower operating temperature. At higher temperatures the effectof thermal losses starts to dominate over the aperture size and wesee that thermal efficiency becomes mainly a function of outlettemperature. It is important to note that since receiver aperturesize is optimized for overall efficiency, the overall efficiency ishigher at 400 �C than at the higher outlet temperatures will beshown in Sec. 4.3.

4.3 Overall Efficiency for Heat and Power Output. Theoverall efficiency (Fig. 10) is the product of optical and thermalefficiencies, as expected it is inversely proportional to the outlettemperature and varying on average from 22% to 26%. Figure 11depicts the thermal power output (kW) collected in the HTF fordifferent temperatures. The total energy varies from 623.1 kWh at400 �C to 529.2 kWh at 700 �C.

Figure 12 depicts the maximum possible mechanical power(exergy rate) of the fluid for varying temperatures, calculatedbased on Carnot efficiency assuming cold reservoir temperature of80 �C. Maximum possible mechanical power is also calculatedfor real-time optimized fluid outlet temperature (Tfo¼ Topt). SeeSec. 4.4.

4.4 Comparison of Control Strategies. The selection of thecontrol strategy is often dictated by the consumer process, mostprocesses require a constant temperature supply from the heatsource. This obvious challenge for solar systems maybe overcomeby some kind of thermal inertia (storage) or hybridization withfossil fuel based sources. Therefore, the solar engineer might havethe choice between constant-temperature-variable-flow controlstrategy, constant-flow-variable-temperature control strategy or avariable-temperature-variable-flow control strategy that maxi-mizes exergy (or other objective function).

While constant flow control is the easiest to implement practi-cally because it requires less hardware and also simpler control,the merits of the other control strategies might justify the change,this has to be assessed on a system level. The objective functionmust be modified to include the consumer process operationparameters.

Figure 13 shows simulation results for the three aforementionedcontrol strategies. It can be seen in (a) that the differencesbetween the three control strategies is small. This is because theconstant temperature and constant flow rate were also selected bymaximizing the objective function.

If the temperature deviates from the optimized constant temper-ature then the mechanical power will be lower (see Fig. 12). Itshould be noted that although the optimal mass flow and outlet

Fig. 9 Receiver thermal efficiency. Average efficiency at 300 �Cis 71%, at 400 �C is 73%, at 500 �C is 71% and at 600 �C is 68%.

Fig. 10 Overall efficiency of the BDOE. Overall efficiency at300 �C is 28%, at 400 �C is 26%, at 500 �C is 25%, and at 600 �C is24%.

Fig. 11 Thermal output of the receiver as function of time andoutlet temperature

Fig. 12 Maximum mechanical power, which is indicative of thesolar-to-electricity efficiency of the BDOE



temperature are constant in this example day, they are expected tochange during the year. Moreover, the small differences amongthe control strategies are affected by receiver design, which, inthis case, has low thermal losses. For less efficient receivers, thereal time optimized temperature control strategy may have a sig-nificant advantage.

5 Conclusions

The overall efficiency of the BDOE is estimated to vary from24% to 28% during the test day. The main losses in the system areoptical losses: cosine loss, reflectivity loss, beam attenuation loss,blocking, and shading loss in addition to light spillage around CRmirrors and around the receiver aperture (intercept factor).

The optical efficiency of the system can be improved by betteradjustment of the heliostats facet canting and also the canting ofthe CR mirrors to avoid spillage around the CR mirrors and thereceiver, respectively.

The beam down optical arrangement suffers certain opticallosses, mainly the CR spillage and reflection losses which are notincurred by a conventional tower receiver system. However, theselosses may be outweighed by the potential to use a cavity typereceiver and possibly a final compound parabolic concentratoroptical element made possible by the relatively narrow angulardistribution (maximum incident angle of 17 deg) of radiation inci-dent on the receiver. Beam down plants have the additionaladvantage of locating the receiver at ground level to reduce instal-lation and operational costs.

Nomenclature

A ¼ receiver aperture area (m2)Cp ¼ specific heat of the HTFðkJkg�1K�1ÞDi ¼ receiver tube inside diameter (m)Do ¼ receiver tube outside diameter (m)F0 ¼ receiver plate efficiency factorFR ¼ heat removal factor

g ¼ gravitational acceleration (m/s2)Gr ¼ Grashof number

G(r, h) ¼ radiation flux map in polar coordinates (kW/m2)�hc ¼ receiver convection heat loss coefficient (kWm�2K�1)hf ¼ convection heat transfer coefficient between HTF and

tube wall (kWm�2K�1)�hr ¼ receiver linearized radiative heat transfer coefficient

(kWm�2K�1)k ¼ thermal conductivity of air (Wm�1K�1)L ¼ receiver characteristic length (m)_m ¼ HTF mass flow rate (kg/s)

Nu ¼ Nusselt numberPr ¼ Prandtl number

_qin ¼ solar flux intercepted by the receiver (kW/m2)_Qrad ¼ radiative heat loss (kW)

_Qu ¼ useful thermal power (kW)R ¼ receiver aperture radius (m)

Re ¼ Reynolds numberTe ¼ ambient air temperature (�C)Tfi ¼ inlet mean fluid temperature (�C)

Tfilm ¼ mean film temperature (�C)Tfo ¼ outlet mean fluid temperature (�C)

Topt ¼ real-time optimized fluid outlet mean temperature (�C)Tmp ¼ mean plate temperature (�C)Tsky ¼ effective sky temperature (�C)UL ¼ overall heat loss coefficient (kWm�2K�1)a ¼ receiver absorptivity in solar spectrumb ¼ volumetric thermal expansion coefficient (K�1)

DT ¼ temperature difference between mean plate temperatureand ambient air temperature (K)

E ¼ receiver longwave emissivityc ¼ receiver intercept factor� ¼ kinematic viscosity of air (m2/s)r ¼ Stefan-Boltzmann constant

(5:670373� 10�8Wm�2K�4)

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